Determine how many solutions exist for the system of equations. ${4x-2y = -8}$ ${-6x+3y = 12}$
Solution: Convert both equations to slope-intercept form: ${4x-2y = -8}$ $4x{-4x} - 2y = -8{-4x}$ $-2y = -8-4x$ $y = 4+2x$ ${y = 2x+4}$ ${-6x+3y = 12}$ $-6x{+6x} + 3y = 12{+6x}$ $3y = 12+6x$ $y = 4+2x$ ${y = 2x+4}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 2x+4}$ ${y = 2x+4}$ Both equations have the same slope and the same y-intercept, which means the lines would completely overlap. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Since any solution of ${4x-2y = -8}$ is also a solution of ${-6x+3y = 12}$, there are infinitely many solutions.